## Abstract

Power battery system is widely used in new energy vehicles. The performance of power battery directly affects the safety of new energy vehicles. At present, the research of battery system safety focuses on specific parameters such as the status of charge (SoC), the state of health (SoH), and so on. However, a single performance evaluation may lead to some defects in system safety. The influences on the power battery system of SoC, SoH, and the status of consistency, which are essential factors in battery system safety were analyzed in this study. In view of the conclusion of the analysis result, the concept of the power battery system performance matrix, which includes the parameters of SoC, SoH, the status of consistency and temperature, was proposed. The state-space expression was studied and used for the expression of the performance matrix. Cyclic charging and discharging experiments of Lithium-ion batteries were carried out, and the result showed that the performance matrix based on state-space could describe the dynamic changes of battery system status. It was sufficient for the estimate of the battery system’s safety status, and it can provide the effective and reliable data resources for the system maintenance.

## 1 Introduction

The power battery system is the core part of new energy vehicles. However, the quality of the system may directly affect the safety, stability, and mileage of the whole vehicle [1]. The performance of the power battery system may change when operating under complex working conditions for a long time. If the corresponding measures are not taken in time according to the battery system performance, the battery system may be damaged, or even an accident may occur. Due to different usage environments, the power battery system presents different characteristics in mechanical deformation, voltage[2], charging/discharging current, temperature [3], internal resistance, etc., which may increase the complexity of battery system performance. From the perspective of safety and convenience, the characteristics of the battery system include three aspects, as follows: (1) The status of charge (SoC) is an important parameter in predicting the remaining energy and the driving range of the vehicle [4]; if the SoC evaluation is incorrect, the battery system may be over charged or over discharged when operating, which would threaten the safety of the battery system. (2) The status of health (SoH) can predict the service life of the battery system. It is an important index for battery system maintenance [5]. Because the aging of the battery may make the performance of the battery system worse and even cause some danger to the new energy vehicle, it is necessary to consider the degree of aging together with the other characteristics. (3) Consistency of the batteries in the system. The battery system is composed of a series connection. Due to the Buckets effect, the evaluation of SoH can reflect the input/output ability of the battery system. It cannot be determined that a single battery or the system prompts the problem of aging. Consistency describes the differences of the cells and reveals the influence of single power battery on the policy [6]. On the other hand, it can determine the existence of the faulty battery in time by the value of consistency. Therefore, the battery system can be more guaranteed to consider the three factors when evaluating the performance of the battery system.

In view of the complexity of the battery system, the current research methods for the evaluation of battery characteristics are as follows: (1) The method of modeling, controlling, and decision-making for the complex system based on “Grey Qualitative Theory” [7], which divides the characteristics of the system into a different degree and classifies the characteristics with discriminant theory. The method can determine the degree of system’s performance. However, it cannot describe the gradual change of system’s performance. For example, the method can express if the battery system is good or not, but it cannot describe the decline process of battery system so that it cannot make a timely response to battery failure. (2) Intelligent identification method [8,9]: it forms the function of self-judgment and prediction of the system by analyzing the characteristics of the battery system. However, the complex calculation and uncertainty require the management system to have high computational power. (3) The system simulation method [10,11]: according to the characteristics of the existing system in the experiment, the method builds the corresponding database to predict the performance of the battery system. However, if there is a large error in the actual operation of the method, it may show the incorrect result due to the idealistic simulation data. (4) Data modeling method [12,13] describes the internal relations and changes law between the input and the output of the system by the mathematical language. Although there are certain prediction accuracy problems for the information collector with a large noise, the method is more popular for the majority of research due to the ease of computation and stability.

There are certain advantages and disadvantages in the methods earlier. But most of them are used to evaluate a specific performance of battery system (such as SoC or SoH), there are significant limitations in evaluating the overall performance of the battery system. Therefore, the methods were unable to describe the battery system performance comprehensively, intuitively, and quickly in operation.

In order to reveal the influence of battery system’s performance, the authors of this paper tried to predict the SoC and consistency of the power battery system by the weighting method, divided the consistency into several levels, and then established the maintenance strategy [14]. The experiment result showed that the technique was effective [15]. At the same time, a series of tests were carried out and the result showed that there is a great influence on the stability of the battery system when the mechanical system changes [16]. However, it did not mean that there is a linear relationship between the consistency of the battery system and the output of energy. The research in Ref. [17] showed that the energy output efficiency of the system could be effectively improved by recombining the battery and limiting the charging/discharging current according to the capacity characteristics of the single battery. Therefore, it is worth dividing the safety of the power battery into different indexes and take different methods to evaluate the indexes.

On the basis mentioned earlier, this paper presents the performance matrix to express the characteristics of battery system’s safety. First, the concept of power battery system performance matrix was proposed, and the state-space expression, which includes the parameters of SoC, SoH, the status of consistency, and the temperature, was used to describe the performance of the battery system. Second, the methodology of performance matrix was studied, and the dynamic indicators were used to describe the changes of SoC, SoH, the status of consistency, and the temperature in operation. Third, a series of Lithium-ion batteries were used in cyclic charging and discharging tests. The result showed that the performance matrix was effective for the status estimate the battery system’s safety. At last, the paper summarized the issues and prospected the study in the future.

## 2 Methodology

### 2.1 Safety Performance Analysis of Power Battery System.

The parameters of material, voltage, internal resistance, temperature, etc., may influence the safety of power battery. However, not all the parameters can be measured directly. Normally, the parameters that can be directly measured in the battery system include voltage, current, and temperature. Other characteristics of power battery, such as resistance, consistency, residual capacity, etc., can be calculated on the basis of these three parameters. In terms of practicality, the parameters related to the safety of power battery include the following four factors: the SoC, the SoH, and the consistence δ, and the temperature T of the battery system.

Therefore, the performance matrix of the power battery system can be shown by the method of state space as follows:
$K=[SoC(t)SoH(q)δ(t)T(t)]$
(1)
where t means the time in operation and q means the qth discharging cycle.

### 2.2 Status of Charge Model Based on Multi-Parameter.

The SoC is the main parameter to predict the remaining energy or driving range of the vehicle in operation, which is directly related to the driving range and the safety of the vehicle. At present, the normal methods to estimate the SoC include voltage method, voltage-current method, and Kalman filter method. However, these methods either have accumulated system errors or have a high computational costs, so that the SoC becomes a problem in the application of the power battery system.

In order to find a more convenient calculation method, some concepts about the operation of power battery should be studied: (1) Open circuit voltage, the terminal voltage of battery when the battery is not charged and discharged; (2) terminal voltage VOP, the potential difference between the positive and negative electrodes of a battery; (3) overpotential ΔVCR, it means the difference between the terminal voltage and the equilibrium potential; and (4) equilibrium potential VEMF means the terminal voltage when the internal electrochemical reaction of the battery is in equilibrium. VEMF can be considered as the open circuit voltage Vop(t0), which is the terminal voltage of battery when the battery core temperature is close to the ambient temperature [1].

The SoC is related to the equilibrium potential (VEMF), and VEMF can be approximated by the terminal voltage Vop and the overpotential ΔVCR of battery at a specific discharging/charging current I and temperature T [1]. The authors have verified the multidimensional relationship between ΔVCR and T, I, and VOP at the moment t that the battery stops charging and discharging by experiments [14]. The expression of SoC can be approximated based on multi-parameter as follows:
$ΔVCR(t)=VOP(t)−VOP(t0)ΔVCR(t)=∑i=0d(∑j=0y(∑g=0xlijgTg(t))Ij(t))VOPi(t)VEMF(t)=VOP(t)−ΔVCR(t)SOC(t)=H(VEMF(t))$
(2)
where lijg is the polynomial coefficients which are related to T, I, and VOP; g, j, and i are the related factors of T, I, and VOP, respectively; d, y, and x are the orders of T, I, and VOP, respectively; the values of them depend on the size of calibration sample. Generally, the higher the orders, the higher the prediction accuracy, and the higher computational costs; t0 is the moment that battery core temperature is the same as the ambient temperature; H is the mapping function between SoC(t) and VEMF(t); it can be obtained by looking up the table about SoC and equilibrium potential and interpolating.

### 2.3 Status of Health Model Based on Capacity Calculation.

The SoH is the parameter to estimate the degree of battery system aging. It can be expressed by comparing the capacity of the system in operation with nominal capacity. However, it can be fully counted in a certain time. So, the function of SoH can be expressed by the Coulomb counting value of last time as follows:
$if(Q(q−1)>Qc)SoH(q)=100%elesSoH(q)=Q(q−1)Qc×100%$
(3)
where q is the number of cycle charging and discharging; Qc means the nominal battery capacity; Q(q−1) in (3) means the measured battery capacity of the (q−1)th cycle. The degree of battery system aging can be described by the calculation of capacity ratio in (3), and SoH can be predicted.

### 2.4 Consistency Evaluation Algorithm.

Because the battery system is composed of series connection, the consistency δ describes the degree of performance difference among the battery system. It reflects the influence of a battery on the whole system when the battery fails, and provides reference for the maintenance. Therefore, the expression of a consistency evaluation algorithm can be shown as
$δ(t)=(σmax−σ(t))σmax×100%σ(t)=∑i=1n(VEMFi(t)−VaEMF(t))2n−1VaEMF(t)=∑i=1nVEMFi(t)n$
(4)
where σ(t) is the standard deviation of all the equilibrium potentials in the time of t; σmax is the reference value of the consistency under the maintenance, which can be obtained by the discharging/charging cycle test. In the paper, the value is 0.2004 V by 2000 times of battery system’s discharging/charging cycle test. VaEMF(t) is the average value of all the battery’s equilibrium potentials in the time of t.

In summary, the performance matrix by the method of state space can describe the details of the battery system in many aspects in real time. The battery safety can be displayed more intuitively by the graphic method and threshold method compared with the numerical description of a single parameter.

### 2.5 Experimental Steps.

The experimental steps were as follows:

• Step 1: The batteries were put into the thermal chamber separately, and the discharging/charging test (Fig. 1) was carried out. The current in the test was set to 1 A, 3 A, 5 A, 10 A, and 15 A separately, and the temperature of the thermal chamber was set to 40 °C, 30 °C, and 20 °C separately. The value of Vop can be obtained by the average value of the batteries’ terminal voltages in the test. And the database of overpotential can be established by the function (2).

• Step 2: According to the database of overpotential, the polynomial coefficients lijg in function (2) can be calculated by the least square method. And the parameter d, y, and x can be set to 2, 3, and 4, respectively, in the test.

• Step 3: The batteries in the test were charged and connected in series. The discharging test with variable current (Fig. 2) was carried out for 1000 times. The performance matrix method proposed was calculated, and the accuracy index γ of the SoC can be estimated as shown as
$γ=σ′Vmax−Vmin×100%σ′=∑i=1n(VEMFc(i)−VEMFr(i))2n$
(5)
where VEMFc(i) is the predicted equilibrium potential of the ith battery; VEMFr(i) is the equilibrium potential in the theory of the ith battery; σ′ is the standard deviation of predicted equilibrium potential error; Vmax is the maximum value of predicted equilibrium potentials; Vmin is the minimum value of predicted equilibrium potentials.
• Step 4: The performance matrix of the batteries in the first discharging test can be compared with the 1000th discharging test by the method earlier. And the degradation of the batteries can be obtained by the comparison data.

## 3 Experiment and Results

### 3.1 Subject Investigated.

In order to verify the proposed method, 8 32650-lithium-ion+ batteries, with a rated capacity of 5Ah, were used (Fig. 3(a)) in the experiment. Neware-CT-4008-5V20A was used in the single-battery test for the calibration of polynomial coefficients in function (2) (Fig. 3(b)), and the batteries test of series connection (Fig. 3(c)) was carried out by the equipment of Neware-CT-4002-50V20A for the experiments of SoH estimation and consistency calculation. DGBELL-BTT-2m3B (Fig. 3(d)) was the thermal chamber.

Fig. 1
Fig. 1
Close modal
Fig. 2
Fig. 2
Close modal
Fig. 3
Fig. 3
Close modal

### 3.2 Experimental Result.

According to Step 1 in Sec. 2.5, the database of overpotential can be established as shown as Fig. 4. The polynomial coefficients in function (2) were calculated as shown as Table 1.

Fig. 4
Fig. 4
Close modal
Table 1

The parameters’ values of lijg

j = 4j = 3j = 2j = 1j = 0
g = 2i = 30.000005−0.0001470.001244−0.0029300.002093
i = 2−0.0000430.001266−0.0106970.024861−0.017953
i = 10.000123−0.0036010.030291−0.114258−0.162960
i = 0−0.0001150.003368−0.0281770.063230−0.046415
g = 1i = 3−0.0003220.009498−0.0808350.188778−0.134067
i = 20.002745−0.0811350.687800−1.5817331.138364
i = 1−0.0077110.228008−1.9239067.4928999.643501
i = 00.007098−0.2101381.764126−3.8963272.865147
g = 0i = 30.004454−0.1330851.144103−2.6578601.711187
i = 2−0.0376231.127267−9.66401922.095518−14.286504
i = 10.104227−3.13154726.767963−114.074423−131.967995
i = 0−0.0941152.840687−24.21966953.256602−33.982809
j = 4j = 3j = 2j = 1j = 0
g = 2i = 30.000005−0.0001470.001244−0.0029300.002093
i = 2−0.0000430.001266−0.0106970.024861−0.017953
i = 10.000123−0.0036010.030291−0.114258−0.162960
i = 0−0.0001150.003368−0.0281770.063230−0.046415
g = 1i = 3−0.0003220.009498−0.0808350.188778−0.134067
i = 20.002745−0.0811350.687800−1.5817331.138364
i = 1−0.0077110.228008−1.9239067.4928999.643501
i = 00.007098−0.2101381.764126−3.8963272.865147
g = 0i = 30.004454−0.1330851.144103−2.6578601.711187
i = 2−0.0376231.127267−9.66401922.095518−14.286504
i = 10.104227−3.13154726.767963−114.074423−131.967995
i = 0−0.0941152.840687−24.21966953.256602−33.982809

In order to verify the effectiveness of the proposed method, voltage-current method, Kalman filter method was used in the comparison test. The data processing was carried out on matlab, and the computer processor was configured as Intel(R) Core(TM) i7-6500 CPU, 8 GB RAM.

The SoC evaluating comparative results of performance matrix K can be shown in Fig. 5. Due to the accumulation of errors, it can be clearly expressed that the proposed method in the paper and Kalman filter method were batter in SoC evaluation than the voltage-current method.

Fig. 5
Fig. 5
Close modal

The SoC evaluation accuracy by (5) and the data processing time of different methods are shown in Table 2. The results were clearly expressed that the proposed method in the paper and the Kalman filter method showed higher accuracy in SoC evaluation. But the data processing time of the voltage-current method was the shortest due to the low algorithm complexity. Considering the accuracy and the calculation time, the method based on multi-parameter was more effective and it can satisfy the accuracy requirement of SoC evaluation.

Table 2

The SoC estimation accuracy in the cyclic discharging comparison test

MethodAccuracy and calculation speedFirst test1000th test
The method proposed in the paperσ′ (V)0.0464610.051694
γ (%)4.0364274.646054
Data processing time (s)a0.76560.7743
Voltage-current methodσ′ (V)0.0686830.073179
γ (%)8.6243699.246530
Data processing time (s)a0.43750.4547
Kalman filter methodσ′ (V)0.0498860.054798
γ (%)5.0214735.248960
Data processing time (s)a1.24681.2342
MethodAccuracy and calculation speedFirst test1000th test
The method proposed in the paperσ′ (V)0.0464610.051694
γ (%)4.0364274.646054
Data processing time (s)a0.76560.7743
Voltage-current methodσ′ (V)0.0686830.073179
γ (%)8.6243699.246530
Data processing time (s)a0.43750.4547
Kalman filter methodσ′ (V)0.0498860.054798
γ (%)5.0214735.248960
Data processing time (s)a1.24681.2342
a

Data processing time is the time required for all data processing in a discharging test.

According to the algorithm of SoH by Eq. (3), the SoH evaluating comparative results of performance matrix K was the same as Fig. 6.

Fig. 6
Fig. 6
Close modal

Due to the algorithm of SoH by Eq. (3), SoH can be considered to 100% when the battery capacity exceeds the nominal capacity (5Ah). Therefore, the aging degree of battery within 250 cycles was 100% in the aging test.

The consistency evaluating comparative results of performance matrix K can be shown in Fig. 7. Due to the potential difference, there are data floats in voltage-current method. The situation may seriously affect the analysis of battery safety. On the contrary, the results of the other two methods were relatively smooth due to the higher accuracy of SoC evaluation.

Fig. 7
Fig. 7
Close modal

Together with the evaluation of SoH, consistency of performance matrix K can indicate whether the degradation of battery performance is due to a battery or the system. So that it was a useful reference for the maintenance.

The temperature results of performance matrix K showed that the battery temperature rise might be related to the discharging cycles in Fig. 8. When the temperature parameter of matrix K rises rapidly, the battery system should be prohibited so as to ensure safety.

Fig. 8
Fig. 8
Close modal

According to the data of matrix K in the comparison test, it can be clearly expressed that the performance of the battery would change when there is some abnormal data of the performance matrix. And it is useful for monitoring and analyzing the safety of batter system.

## 4 Conclusion

The paper focused on the performance of the power battery in use, which includes the value of SoC, SoH, the status of consistency, and the temperature. The concept of the power battery system performance matrix based on state-space was proposed. And the expression of the performance matrix was discussed. The main work was as follows:

1. The relationship between the equilibrium potential and the terminal voltage, current, temperature, and SoC was analyzed. The evaluation model of SoC based on multi-parameter was proposed. The accuracy comparison test proved that the model is effective.

2. The evaluation method of SoH was discussed and the evaluation model based on the coulomb counting value of the last complete discharging process was expressed. The cyclic discharging test verified that it is helpful to the analysis of battery security performance together with other parameters in the performance matrix K.

3. According to the standard deviation of all the battery’s equilibrium potentials in use, the expression of battery system’s consistency was established. And the degree of equilibrium potentials’ difference among the battery system was proposed and the test proved that the performance matrix could indicate whether the degradation of battery performance is due to a battery or the system.

4. The discharging test with the variable current was carried out for 1000 times, and the result of performance matrix proved the method can describe the changes in different battery characteristics and is beneficial to the maintenance and evaluation of the battery system.

However, several further research works are as follows:

1. There are some noises in the performance curve due to the influence of the test data. It is necessary to take measures to filter the noises.

2. SoC, SoH, consistency of the battery system, and the temperature of the system were considered in the paper. However, there are other factors that may affect the safety of the battery system. They can also be taken into the performance matrix. How to build the expression models of the factors by analyzing the characteristics of the factors is an important work in the future.

## Acknowledgment

The authors would like to thank the support of the Thermal Safety Technology Key Laboratory of New Energy vehicle’s Power System and the Guangdong Key Laboratory of Intelligent Transportation System (Grant No. 202005004), the Natural Science Foundation of Guangdong Province, China (Grant No. 2018A030313753), the Innovative Research Team of System Safety of Battery Pack for New Power Vehicle, Guangdong Province, China, and the Guangzhou People's Livelihood Science and Technology Project (Grant No. 201803030041), China.

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